As part of a study of corporate employees, the Director of...

As part of a study of corporate employees, the Director of Human Resources for PNC, Inc. wants to compare the distance traveled to work by employees at their office in downtown Cincinnati with the distance for those in downtown Pittsburgh. A sample of 35 Cincinnati employees showed they travel a mean of 370 miles per month, with a standard deviation of 30 miles per month. A sample of 40 Pittsburgh employees showed they travel a mean of 380 miles per month, with a standard deviation of 26 miles per month. At the .05 significance level, is there a difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees? Use the five-step hypothesis-testing procedure.

Key Concepts

Standard Error and Degrees of Freedom

The standard error quantifies the variability of the difference between sample means and is used to standardize the test statistic. In a two-sample test, it incorporates the variability from both samples. The degrees of freedom determine the appropriate distribution for the test statistic, affecting the critical values and p-value computations. They are calculated based on the sample sizes and are particularly important when using the t-distribution for smaller samples or when variances are estimated from the data.

Test Statistic

The test statistic is a standardized value computed from sample data during a hypothesis test. It measures the degree to which the sample results diverge from what is expected under the null hypothesis. For comparing two means, the test statistic typically involves the difference between sample means divided by the estimated standard error of that difference, which follows a t-distribution or normal distribution depending on sample sizes and variance assumptions.

Significance Level

The significance level, denoted as alpha (?), represents the threshold for rejecting the null hypothesis. It is the probability of making a Type I error, which is the error of rejecting a true null hypothesis. In this case, a 0.05 significance level is used, meaning there is a 5% risk of wrongly concluding that a difference exists between the two groups.

Two-Sample Comparison of Means

This concept involves comparing the means of two independent samples to determine if there is a statistically significant difference between them. It is widely used in situations where two groups are subjected to different conditions or treatments, and researchers want to understand if the observed differences in sample means reflect true differences in the populations.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. The method involves stating a null hypothesis, which is the default assumption of no effect or difference, and an alternative hypothesis, which represents the claim to be tested. The process involves collecting data, calculating a test statistic, and comparing it to a critical value or using a p-value to determine whether to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) states that there is no difference in the population means, while the alternative hypothesis (Ha) posits that there is a difference. Formulating these hypotheses correctly is critical as they offer a clear framework for statistical testing and decision-making based on the evidence provided by the sample data.

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